Jet schemes and invariant theory

Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin

doi:10.5802/aif.2996

Jet schemes and invariant theory
[Schémas des jets et théorie des invariants]

Linshaw, Andrew R. ; Schwarz, Gerald W. ; Song, Bailin

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2571-2599.

Résumé
Abstract

Soient $G$ un groupe réductif complexe et $V$ un $G$ -module. Alors $G_{m}$ , le schéma des jets d’ordre $m$ de $G$ , opère dans $V_{m}$ , le schéma des jets d’ordre $m$ de $V$ , pour tout $m \geq 0$ . Nous nous intéressons à l’anneau des invariants $𝒪 {(V_{m})}^{G_{m}}$ et au morphisme $p_{m}^{*} : 𝒪 ({(V / / G)}_{m}) \to 𝒪 {(V_{m})}^{G_{m}}$ induit par le morphisme du quotient catégorique $p : V \to V / / G$ : ce morphisme est-il un isomorphisme, surjectif, ou non ? En utilisant le théorème du slice de Luna, nous obtenons des critères pour que $p_{m}^{*}$ soit un isomorphisme pour tout $m$ . Nous montrons que c’est bien le cas lorsque $G = {SL}_{n}$ , ${GL}_{n}$ , ${SO}_{n}$ , ou ${Sp}_{2 n}$ et $V$ est un somme directe de copies du module standard et de son dual, pourvu que $V / / G$ soit lisse ou une intersection complète. Nous classifions toutes les représentations de $ℂ^{*}$ telles que $p_{\infty}^{*}$ soit surjectif ou un isomorphisme. Enfin, nous donnons des exemples où $p_{m}^{*}$ est surjectif pour $m = \infty$ mais non surjectif pour $m$ fini, et d’autres exemples où $p_{m}^{*}$ est surjectif mais non injectif.

Let $G$ be a complex reductive group and $V$ a $G$ -module. Then the $m$ th jet scheme $G_{m}$ acts on the $m$ th jet scheme $V_{m}$ for all $m \geq 0$ . We are interested in the invariant ring $𝒪 {(V_{m})}^{G_{m}}$ and whether the map $p_{m}^{*} : 𝒪 ({(V / / G)}_{m}) \to 𝒪 {(V_{m})}^{G_{m}}$ induced by the categorical quotient map $p : V \to V / / G$ is an isomorphism, surjective, or neither. Using Luna’s slice theorem, we give criteria for $p_{m}^{*}$ to be an isomorphism for all $m$ , and we prove this when $G = {SL}_{n}$ , ${GL}_{n}$ , ${SO}_{n}$ , or ${Sp}_{2 n}$ and $V$ is a sum of copies of the standard module and its dual, such that $V / / G$ is smooth or a complete intersection. We classify all representations of $ℂ^{*}$ for which $p_{\infty}^{*}$ is surjective or an isomorphism. Finally, we give examples where $p_{m}^{*}$ is surjective for $m = \infty$ but not for finite $m$ , and where it is surjective but not injective.

Reçu le : 2014-10-03
Accepté le : 2015-02-24
Publié le : 2015-12-08

Zbl 1342.13009

DOI : https://doi.org/10.5802/aif.2996
Classification : 13A50, 14L24, 14L30
Mots clés : schémas des jets, théorie classique des invariants

@article{AIF_2015__65_6_2571_0,
     author = {Linshaw, Andrew R. and Schwarz, Gerald W. and Song, Bailin},
     title = {Jet schemes and invariant theory},
     journal = {Annales de l'Institut Fourier},
     pages = {2571--2599},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2996},
     zbl = {1342.13009},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2996/}
}

Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin. Jet schemes and invariant theory. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2571-2599. doi : 10.5802/aif.2996. https://aif.centre-mersenne.org/articles/10.5802/aif.2996/

Bibliographie

[1] Batyrev, Victor V. Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, pp. 1-32 | MR 1672108 | Zbl 0963.14015

[2] Beilinson, Alexander; Drinfeld, Vladimir Chiral algebras, American Mathematical Society Colloquium Publications, Tome 51, American Mathematical Society, Providence, RI, 2004, vi+375 pages | MR 2058353 | Zbl 1138.17300

[3] Borcherds, Richard E. Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A., Tome 83 (1986) no. 10, pp. 3068-3071 | Article | MR 843307 | Zbl 0613.17012

[4] Boutot, Jean-François Singularités rationnelles et quotients par les groupes réductifs, Invent. Math., Tome 88 (1987) no. 1, pp. 65-68 | Article | MR 877006 | Zbl 0619.14029

[5] Craw, Alastair An introduction to motivic integration, Strings and geometry (Clay Math. Proc.) Tome 3, Amer. Math. Soc., Providence, RI, 2004, pp. 203-225 | MR 2103724 | Zbl 1156.14307

[6] Denef, Jan; Loeser, François Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., Tome 135 (1999) no. 1, pp. 201-232 | Article | MR 1664700 | Zbl 0928.14004

[7] Denef, Jan; Loeser, François Geometry on arc spaces of algebraic varieties, European Congress of Mathematics, Vol. I (Barcelona, 2000) (Progr. Math.) Tome 201, Birkhäuser, Basel, 2001, pp. 327-348 | MR 1905328 | Zbl 1079.14003

[8] Eck, David Invariants of $k$ -jet actions, Houston J. Math., Tome 10 (1984) no. 2, pp. 159-168 | MR 744898 | Zbl 0568.14007

[9] Ein, Lawrence; Mustaţă, Mircea Jet schemes and singularities, Algebraic geometry—Seattle 2005. Part 2 (Proc. Sympos. Pure Math.) Tome 80, Amer. Math. Soc., Providence, RI, 2009, pp. 505-546 | Article | MR 2483946 | Zbl 1181.14019

[10] Frenkel, Edward; Ben-Zvi, David Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, Tome 88, American Mathematical Society, Providence, RI, 2001, xii+348 pages | Article | MR 1849359 | Zbl 0981.17022

[11] Frenkel, Igor; Lepowsky, James; Meurman, Arne Vertex operator algebras and the Monster, Pure and Applied Mathematics, Tome 134, Academic Press, Inc., Boston, MA, 1988, liv+508 pages | MR 996026 | Zbl 0674.17001

[12] Ishii, Shihoko; Kollár, János The Nash problem on arc families of singularities, Duke Math. J., Tome 120 (2003) no. 3, pp. 601-620 | Article | MR 2030097 | Zbl 1052.14011

[13] Kac, Victor Vertex algebras for beginners, University Lecture Series, Tome 10, American Mathematical Society, Providence, RI, 1998, vi+201 pages | MR 1651389 | Zbl 0924.17023

[14] Kolchin, E. R. Differential algebra and algebraic groups, Academic Press, New York-London, 1973, xviii+446 pages (Pure and Applied Mathematics, Vol. 54) | MR 568864 | Zbl 0264.12102

[15] Kontsevich, M. String cohomology, 1995 (Lecture at Orsay)

[16] Kraft, Hanspeter Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984, x+308 pages | Article | MR 768181 | Zbl 0569.14003

[17] Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin Arc spaces and the vertex algebra commutant problem, Adv. Math., Tome 277 (2015), pp. 338-364 | Article | MR 3336089

[18] Looijenga, Eduard Motivic measures, Astérisque (2002) no. 276, pp. 267-297 (Séminaire Bourbaki, Vol. 1999/2000) | Numdam | MR 1886763 | Zbl 0996.14011

[19] Luna, Domingo Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, p. 81-105. Bull. Soc. Math. France, Paris, Mémoire 33 | Numdam | MR 342523 | Zbl 0286.14014

[20] Malikov, Fyodor; Schechtman, Vadim Chiral de Rham complex. II, Differential topology, infinite-dimensional Lie algebras, and applications (Amer. Math. Soc. Transl. Ser. 2) Tome 194, Amer. Math. Soc., Providence, RI, 1999, pp. 149-188 | MR 1729362 | Zbl 0999.17037

[21] Malikov, Fyodor; Schechtman, Vadim; Vaintrob, Arkady Chiral de Rham complex, Comm. Math. Phys., Tome 204 (1999) no. 2, pp. 439-473 | Article | MR 1704283 | Zbl 0952.14013

[22] Mustaţă, Mircea Jet schemes of locally complete intersection canonical singularities, Invent. Math., Tome 145 (2001) no. 3, pp. 397-424 (With an appendix by David Eisenbud and Edward Frenkel) | Article | MR 1856396 | Zbl 1091.14004

[23] Nash, John F. Jr. Arc structure of singularities, Duke Math. J., Tome 81 (1995) no. 1, p. 31-38 (1996) (A celebration of John F. Nash, Jr.) | Article | MR 1381967 | Zbl 0880.14010

[24] Schwarz, Gerald W. Representations of simple Lie groups with regular rings of invariants, Invent. Math., Tome 49 (1978) no. 2, pp. 167-191 | Article | MR 511189 | Zbl 0391.20032

[25] Song, Bailin The global sections of the chiral de Rham complex on a Kummer surface (http://arxiv.org/abs/1312.7386)

[26] Veys, Willem Arc spaces, motivic integration and stringy invariants, Singularity theory and its applications (Adv. Stud. Pure Math.) Tome 43, Math. Soc. Japan, Tokyo, 2006, pp. 529-572 | MR 2325153 | Zbl 1127.14004

[27] Weyl, Hermann The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939, xii+302 pages | MR 1488158 | Zbl 1024.20502

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT